On reductio ad absurdum

Bill asks for an example of proof (of non-existence) by RAA that illustrates my contention that a subject-predicate assertion makes no existential claim.   Two arguments spring immediately to mind:  the mathematical proof that there can be no rational square root of two, and the astronomical argument that there can be no planet Vulcan.  These seem to me to have such similar structure that we  can treat them together.

Suppose, (H) there is a rational square root of two.  Then there is a pair of integers, p and q, that satisfy the following conditions: (a) p and q have no common factor apart from 1, and (b) p2=2q2.  Condition (a) is the usual demand that a rational number p/q be expressed as a ratio of integers in lowest terms.  Condition (b) expresses the requirement that (p/q)2=2.  On the surface this looks unproblematic but a little digging reveals that (a) and (b) cannot both be satisfied.  From (b) it follows that p is even, since only the squares of even numbers are even, and (b) tells us that p2 is even.  So p=2r, say, and (b) can be written as 4r2=2q2, or (b') q2=2r2.  But from (b') we infer that q is even, by the same argument that showed us p was even.  Hence both p and q are even.  That is, they have a common factor of 2, contradicting condition (a).  So from hypothesis H falsehood follows.  Ergo H itself is false and there is no rational square root of two.

Suppose, (H) there is a planet Vulcan.  Then (a) Vulcan is large enough to account by Newtonian gravitational principles for the observed anomalies in the perihelion of Mercury, and (b) Vulcan is small enough to be unobserved from Earth.  It follows from further calculations and plausible assumptions as to the mean density of the planet that the diameter of Vulcan is greater than some distance D, to satisfy (a), and also smaller than D, to satisfy (b).  Again, from hypothesis H falsehood follows. Hence H itself is false and there is no planet Vulcan.

What interests me in these arguments is the role played by the names p, q, r, and Vulcan.  It seems that these names are introduced on the strength of an existence assertion alone, and indeed an existence assertion which turns out to be false.  This however does not detract from the meaningfulness of the names.  For me this is a knock-down argument against any direct reference theory of proper names and makes such theories untenable.  Furthermore, a statement such as p is even that occurs within the argument cannot of itself include the assertion that p exists, or that something exists that satisfies the defining predicate associated with the name p.  Rather, there is in the background, as it were, an existence assertion on which p is even depends for its meaningfulness, but p is even does not of itself make this claim.  And as I find it hard to draw semantic distinctions between sentences occurring within explicit hypotheses and sentences occurring 'on the surface', I'm inclined to deny that a sentence like Max is black is making an existence claim.  Sentences ascribing predicates to named subjects should be man enough to own up to their own dependency on earlier sentences making existential claims.  The adolescent chest-beating that 'the thing I'm about really exists, I tell you!'  convinces nobody.

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